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Math Help - measurable function

  1. #1
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    measurable function

    Prove that if f is a real function on a measurable space X such that {x:f(x) is greater than or equal to r} is measurable for every rational r, then f is measurable.

    I am pretty certain I have to use the density of the rationals, but I don't see how.
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  2. #2
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    Quote Originally Posted by robeuler View Post
    Prove that if f is a real function on a measurable space X such that {x:f(x) is greater than or equal to r} is measurable for every rational r, then f is measurable.

    I am pretty certain I have to use the density of the rationals, but I don't see how.
    If \{x\,:\, f(x) \geq r\} is measurable for every r\in \mathbb{Q}, it follows that \{x\,:\, f(x) > s\}=\bigcup_{r\in \mathbb{Q}, r>s}\{x\,:\, f(x)\geq r\} is measurable for every s\in \mathbb{R}, because a countable union of measurable sets is measurable.
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